Question

Sketch these Cartesian products on the \(x-y\) plane \(\mathbb{R}^{2}\) (or \(\mathbb{R}^{3}\) for the last two). $$ [1,2] \times\\{1,1.5,2\\} $$

Short answer

The points that form the Cartesian product are: [(1, 1), (1.5, 1), (2, 1), (1, 1.5), (1.5, 1.5), (2, 1.5), (1, 2), (1.5, 2), (2, 2)].

Step-by-step solution

Understand the Cartesian Product

A Cartesian product of two sets A and B, denoted by \(A \times B\), is the set of all ordered pairs \((a, b)\) such that \(a \in A\) and \(b \in B\). In this case, set A is the interval [1, 2] and set B is \{1, 1.5, 2\}. Therefore \(A \times B\) represents the set of all ordered pairs \((a, b)\) where \(a\) is an element from [1, 2] and \(b\) is an element from \{1, 1.5, 2\}.

Create the Cartesian Product

Write out the elements of the Cartesian product \(A \times B\). This results in the following 3 pairs: \((1, 1)\), \((1.5, 1)\), \((2, 1)\), \((1, 1.5)\), \((1.5, 1.5)\), \((2, 1.5)\), \((1, 2)\), \((1.5, 2)\), and \((2, 2)\). Then, plot these ordered pairs using an x-y coordinate system.

Plot the Cartesian Product

Draw a 2-dimensional graph with labeled axes. The x-axis can represent set A, and the y-axis can represent set B. Now, plot each ordered pair (a, b) as a point in the plane with x-coordinate a and y-coordinate b.